Structure constant

Applications

• A Lie group is abelian exactly when all structure constants are 0.
• A Lie group is real exactly when its structure constants are real.
• The structure constants are completely anti-symmetric in all indices if and only if the Lie algebra is a direct sum of simplecompact Lie algebras.
• A nilpotent Lie group admits a lattice if and only if its Lie algebra admits a basis with rational structure constants: this is Malcev's criterion. Not all nilpotent Lie groups admit lattices; for more details, see also Raghunathan.
• In quantum chromodynamics, the symbol ${\displaystyle G_{\mu \nu }^{a}\,}$ represents the gauge covariant gluon field strength tensor, analogous to the electromagnetic field strength tensor, F, in quantum electrodynamics. It is given by: $${\displaystyle G_{\mu \nu }^{a}=\partial _{\mu }{\mathcal {A}}_{\nu }^{a}-\partial _{\nu }{\mathcal {A}}_{\mu }^{a}+gf^{abc}{\mathcal {A}}_{\mu }^{b}{\mathcal {A}}_{\nu }^{c}\,,}$$ where f_abc are the structure constants of SU(3). Note that the rules to push-up or pull-down the a, b, or c indexes are trivial, (+,... +), so that f = f_abc = f
  _bc whereas for the μ or ν indexes one has the non-trivial relativistic rules, corresponding e.g. to the metric signature (+ − − −).